Newspace parameters
| Level: | \( N \) | \(=\) | \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 936.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(150.119255345\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.54905.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 60x + 20 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 312) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{3} - x^{2} - 60x + 20 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + \nu - 40 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{2} + 7\nu + 38 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + 7\beta _1 + 159 ) / 4 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 0 | 0 | −70.2033 | 0 | −64.2928 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 15.2101 | 0 | −66.9296 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 62.9932 | 0 | 85.2224 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(13\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 936.6.a.h | 3 | |
| 3.b | odd | 2 | 1 | 312.6.a.e | ✓ | 3 | |
| 12.b | even | 2 | 1 | 624.6.a.q | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 312.6.a.e | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 624.6.a.q | 3 | 12.b | even | 2 | 1 | ||
| 936.6.a.h | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 8T_{5}^{2} - 4532T_{5} + 67264 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(936))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} - 8 T^{2} + \cdots + 67264 \)
|
| $7$ |
\( T^{3} + 46 T^{2} + \cdots - 366720 \)
|
| $11$ |
\( T^{3} - 812 T^{2} + \cdots - 4250112 \)
|
| $13$ |
\( (T - 169)^{3} \)
|
| $17$ |
\( T^{3} + \cdots + 1503781672 \)
|
| $19$ |
\( T^{3} + \cdots - 1634520352 \)
|
| $23$ |
\( T^{3} + \cdots + 2376327168 \)
|
| $29$ |
\( T^{3} + \cdots + 9842619912 \)
|
| $31$ |
\( T^{3} + \cdots - 4962140928 \)
|
| $37$ |
\( T^{3} + \cdots - 415746183592 \)
|
| $41$ |
\( T^{3} + \cdots + 3603001534128 \)
|
| $43$ |
\( T^{3} + \cdots + 1801857514176 \)
|
| $47$ |
\( T^{3} + \cdots + 468694506496 \)
|
| $53$ |
\( T^{3} + \cdots + 26283641596248 \)
|
| $59$ |
\( T^{3} + \cdots + 3061733904640 \)
|
| $61$ |
\( T^{3} + \cdots + 16857050073400 \)
|
| $67$ |
\( T^{3} + \cdots - 99579619714976 \)
|
| $71$ |
\( T^{3} + \cdots + 16016399388288 \)
|
| $73$ |
\( T^{3} + \cdots + 137295199656 \)
|
| $79$ |
\( T^{3} + \cdots - 89073162215424 \)
|
| $83$ |
\( T^{3} + \cdots + 17266708920064 \)
|
| $89$ |
\( T^{3} + \cdots + 68850769971840 \)
|
| $97$ |
\( T^{3} + \cdots + 178791904801800 \)
|
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